Games on Graphs

Games on Graphs

Rob Axtell

Examples

• Abstract graphs: Coordination in fixed social nets (w/ J Epstein)

• Empirical graphs: Peer effects in fixed social networks w/addiction

• Dynamic graphs: Crime waves in endogenously changing networks (w/ George Tita)

Coordination in Transient Social Networks:A Model of the Timing of Retirement

Coordination in Transient Social Networks:A Model of the Timing of Retirement

Joint work with J. Epstein

In Behavioral Dimensions of Retirement Economics, H. Aaron, editor, Brookings Institution Press and Russell Sage Foundation

The DataThe Data

The DataThe Data

The DataThe Data

Coordination Gamein Social Networks

Coordination Gamein Social Networks

A agents, each has a social network, Ni

Coordination Gamein Social Networks

Coordination Gamein Social Networks

A agents, each has a social network, Ni

x {working, retired}A is the state of the society

Coordination Gamein Social Networks

Coordination Gamein Social Networks

Ui x( ) = u xi ,xj( )j∈Ni

∑

A agents, each has a social network, Ni

x {working, retired}A is the state of the society

work retire

work w , w 0, 0

retire 0, 0 r, r

Coordination Gamein Social Networks

Coordination Gamein Social Networks

Ui x( ) = u xi ,xj( )j∈Ni

∑

A agents, each has a social network, Ni

x {working, retired}A is the state of the society

work retire

work w , w 0, 0

retire 0, 0 r, r

Coordination Gamein Social Networks

Coordination Gamein Social Networks

Ui x( ) = u xi ,xj( )j∈Ni

∑

A agents, each has a social network, Ni

x {working, retired}A is the state of the society

τ =w

w + r

Base Case ParameterizationBase Case Parameterization

Parameter Value

Agents/cohort, C 100

rational agents 10%

imitative agents 85%

imitation threshold, τ 0.50

, social network size S [10, 25]U

, network age extent E [-5, 5]U

random agents 5%

p 0.50

Typical Time Series:Rapid Establishment of Age 65 Norm

Typical Time Series:Rapid Establishment of Age 65 Norm

5 10 15 20 Time

0.2

0.4

0.6

0.8

1FractionRetired

Typical Time Series:Nonmonotonic Path to Age 65 Norm

Typical Time Series:Nonmonotonic Path to Age 65 Norm

100 200 300 400 500Time

0.2

0.4

0.6

0.8

1FractionRetired

Establishment of Age 65 RetirementNorm as a Function of Population Types

Establishment of Age 65 RetirementNorm as a Function of Population Types

0 5 10 15 20 25% Rational

10

20

50

100

200

500

1000

2000

Transition Time

0% Random5%

10%

Establishment of Age 65 RetirementNorm as a Function of τ

Establishment of Age 65 RetirementNorm as a Function of τ

0.05 0.1 0.15 0.2 0.25ThresholdStd. Dev.

20

40

60

80

100TransitionTime

Establishment of Age 65 RetirementNorm as a Function of Network Size

Establishment of Age 65 RetirementNorm as a Function of Network Size

5 10 15 20 25 30 35 40MeanNetworkSize

1020

50

100

200

5001000

2000

TransitionTime

Establishment of Age 65 Retirement Normas a Function of Variance in Network Size

Establishment of Age 65 Retirement Normas a Function of Variance in Network Size

2 4 6 8 10NetworkSizeStd. Dev.

10

15

20

30

TransitionTime

10 20 30 40 50 60 70 80Max. NetworkSize

10

100

1000

TransitionTime

Establishment of Age 65 Retirement Normas a Function of S, |N| ~ U[10, S]

Establishment of Age 65 Retirement Normas a Function of S, |N| ~ U[10, S]

2 4 6 8 10Extent

10

152030

5070100150200

300Transition Time

5% Rational10% Rational

Establishment of Age 65 Retirement Normas a Function of the Extent of Social Networks

Establishment of Age 65 Retirement Normas a Function of the Extent of Social Networks

0 2 4 6 8 10 12 14% Rational

10

1520

30

5070100

150200

Transition Time

Establishment of Age 62 Retirement Normas a Function of the Extent of Social Networks

Establishment of Age 62 Retirement Normas a Function of the Extent of Social Networks

0.05 0.1 0.15 0.2 0.25Coupling

20

50

100

200

500

1000Transition Time

Sub- population withrational agents

Sub- population withoutrational agents

Establishment of Age 65 Retirement Normas a Function of the Coupling Between Groups

Establishment of Age 65 Retirement Normas a Function of the Coupling Between Groups

Effect of Interaction TopologyEffect of Interaction Topology

• Random graphs

Effect of Interaction TopologyEffect of Interaction Topology

• Random graphs

• Regular graphs (e.g., lattices)

Effect of Interaction TopologyEffect of Interaction Topology

• Random graphs

• Regular graphs (e.g., lattices)

• ‘Small-world’ graphs

New ParameterizationNew Parameterization

Parameter Value

Agents/cohort, C 100

rational agents 10%

imitative agents 85%

imitation threshold, τ 0.50

, social network size S 24 , network age extent E [-5, 5]U

random agents 5%

p 0.50

0 5 10 15 20% Rational

10

20

50

100

200

500

1000

2000

Transition Time

Random graph

Small world

p = 10% p = 25%

Lattice

Comparison of Random Graph, Latticeand Small World Social Networks

(Network size = 24)

Comparison of Random Graph, Latticeand Small World Social Networks

(Network size = 24)

An Empirical Agent Model of Smoking with Peer Effects

• Population of Agents– Arranged in classrooms

– Each agent has a social network

• Agents are Heterogeneous– Distribution of initial thresholds, τ: fraction (f) of an

agent’s social network who must smoke before an agent adopts smoking

– Behavioral rule: If f > τ then smoke, else don’t (or quit)

– Threshold of 1 means non-smoker, 0 first adopter

Agent Behavior• Agents update their behavior periodically

• Smoking reduces threshold:– Decreases with amount smoked– Decreases with intensity of smoking

τ

amount of smoking

τ0

Visualization

Cohorts

Threshold 1 agent (never smokes)

Intermediate threshold agent

Threshold 0 agent (always smokes)

Non social network agent

Smoker

< Run Model>

Typical Output: Smoking Time Series

Lesson: Significant temporal variations in aggregate data;non-equilibrium, non-monotonic

Estimating the Peer Effects

Real world

Estimating the Peer Effects

Real worldStandard specification

Extent of peer effects

Estimation ofmis-specifiedmodel

Estimating the Peer Effects

Real worldStandard specification

Extent of peer effects

Estimation ofmis-specifiedmodel

Agent-Based Model

Estimation ofmis-specified modelwith ‘synthetic’ data

Estimation ofagent model

Conventional (Mis-)Specification

€

yi =+1−1

⎧ ⎨ ⎩

€

yi* =xib + ρijy j +εi

j≠i∑

€

yi*

( )t+1

=xitb + ρijy j

t +εit

j≠i∑

€

θ ≡ b, ρ( )) θ =aρgmax

θlθ( )

€

lθ( ) = Q y j x j ,θ( )−log εxp Q η j x j ,θ( )∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥j=1

N

∑

Typical Results

• Nakajima (2003)– 2000 National Youth Tobacco Survey (NYTS)

• 35K students

• Grades 6-12

• 324 high schools

– Peer effects estimated:• ρff = 0.89

• ρfm = ρmf = 0.48

• ρmm = 0.94

• Krauth…

Crack, Gangs, Guns and Homicide:A Computational Agent Model

George TitaUC Irvine

Rob AxtellBrookings

Drug-Related Homicide in Largest 237 U.S. Cities, mid 1980s to Present

(Blumstein, Cork, Cohen and Tita)

• Innovation in narcotics: crack cocaine

• Emergence of gangs

• Adoption of guns

• Rise of gun violence and homicide

• Diffusion of non-drug gun homicide

An Agent Model

• The problem domain well-suited to agent modeling because:– Heterogeneous actors– Social interactions– Purposive but not hyper-rational behavior– Non-equilibrium dynamics

• Preliminary results to be shown

Basic Features of Model

• Payoffs depend on context (to be described)• Population of drug sellers who interact with

one another through social networks (random graph, lattice and small world)

• Agents heterogeneous wrt age, network• Agents removed by incarceration (fixed

rate), becoming too old (age 40), or death (proportional to amount of gun toting)

Payoffs to Selling Drugs

No guns GunsNo guns 3, 3 0, 4-G

Guns 4-G, 0 1, 1

where G is the price of buying+owning+using a gun

If G is large, this is the assurance (stag hunt) gameIf G is small, this is prisoner’s dilemma

Pre-Crack Era

No gun GunNo gun 3, 3 0, 2

Gun 2, 0 1, 1

Payoffs low (relatively), price of guns (relatively) high

Two Nash equilibria in the assurance game, much like acoordination game; ‘no gun’ equilibrium is Pareto efficient

Crack Era

Payoffs high (relatively), price of guns (relatively) low

‘Gun toting’ is dominant strategy in prisoner’s dilemma,although ‘no gun’ outcome Pareto dominates the Nash outcome

No gun GunNo gun 3, 3 0, 4

Gun 4, 0 1, 1

Economic Emergence of Gangs

‘Gun toting’ is dominant strategy for a gang of size N > G

No guns GunsNo guns 3N, 3N 0, 4N-G

Guns 4N-G , 0 N, N

Widespread ‘gun toting’ leads to drug-related homicide

Drug-Related Homicide

• Goal: explain peaks and troughs in drug homicide rates (e.g., Watts: 30<->120/100K)

• Postulate homicide rate proportional to rate of gun ownership

• Homicide is one more way an agent can be removed from the population (in addition to being incarcerated and becoming too old)

• This can lead to oscillatory homicide rate dynamics

Typical Model Output

Annual drug-related homicides

Year

Summary

• Simple model:– Adaptive agents– Social networks

• Preliminary results:– Multiple regimes, sensitive to network structure– Qualitative plausibility

• Much future work to do– Comments welcome